Decay Rate for decomposing bacteria-word problem

Question

The rate that a chemical decomposes is increasing 1.9 percent continuously. What's the actual hourly percent increase?

I feel like these questions are too vague, is there a formula I can use that I've been spacing?

Answer 1

This is essentially the difference between a compounding process and a continuous process:

Compound interest can be calculated using the formula A = P (1 + r/n) ^(nt), entering into it the initial principal amount (P), annual interest rate (r as decimal), time factor (t) and the number of compound periods (n). In the continuous process we have (1+ r/n)^(nt) becomes e^rt

Here t is in seconds and rt is 0.019/secx t sec usually we measure rates of reactions in terms of concentration changes per second an hour is 3600 seconds so we are looking at e^68.4 as the multiplier of our decomposed reactant between t = 0 seconds and t = 3600 seconds

I agree a bit, as I would like continuous defined a bit better as 1.9 percent...we have processes that can be observed to the femto second or less....so that is vague. I think the point is to look at first order decomposition reactions where reactant concentration measured goes as [R] = [R_o] e ^-Kt t in seconds and R in concentration units and K a rate constant in reciprocal seconds.

Answer 2

You don't need additional information. Use the formula for continuous compounding, A = Pe^rt with e equal to Euler's number 2.718281828.... On a TI graphing calculator, use the e^x key about the LN key and enter e to the first power. (Rounding the e value will cause inaccuracy when you are working with very small numbers.) You can use P as 1 since no initial chemical amount is given. For the rate, r, use 0.019. For time, t, use 1 year divided by the number of hours per year, 365 x 24 = 8760. So, Pe^rt is e^(0.019*1/8760) = 1.000002169. Since we are using 1 as the initial amount, the increase is .000002169 in one hour, or 0.0002169%. You can test this answer by substituting it into the equation for hourly compounding. So 1.000002169^(365*24) = 1.01918, matching the yearly increase of 1.9% given in the problem. I hope this is helpful!